q-Painlev e equations as periodic reductions of integrable ...cormerod/talks/WavesOrmerod2013.pdf · integrable lattice equations Christopher M. Ormerod in collaboration with P H

q-Painleve equations as periodic reductions ofintegrable lattice equations

Christopher M. Ormerodin collaboration with

P H van der Kamp and G R W Quispel

La Trobe University

23rd of March, 2013

Christopher M. Ormerod in collaboration with P H van der Kamp and G R W Quispel La Trobe University ()q-Painleve equations as periodic reductions of integrable lattice equations23rd of March, 2013 1 / 21

Abstract

We describe a direct method to obtain Lax pairs for traveling wave(periodic) reductions of a class of discrete non-autonomous solitonequations. We will derive a q-analogue of the sixth Painleve equationas reductions of both the discrete modified Korteweg-de Vries equationand the discrete Schwarzian Korteweg-de Vries equation.


We consider lattice equations of the form

Q(wl ,m,wl+1,m,wl ,m+1,wl+1,m+1;αl , βm) = 0 (1)

where Q is multilinear and multidimensionally consistent and αl and βmare functions of l and m respectively. Given a staircase of initial values,one evolves the system by imposing (1) on each square in Z2.

With this initial data, we may find wl ,m for all (l ,m) ∈ Z2.






















The two examples we will consider are the discrete modified Korteweg-deVries equation

αl(wl ,mwl+1,m − wl ,m+1wl+1,m+1)

− βm(wl ,mwl ,m+1 − wl+1,mwl+1,m+1) = 0,

and the discrete Schwarzian Korteweg-de Vries equation

αl

(1

wl ,m+1 − wl+1,m+1+

1

wl+1,m − wl ,m

)= βm

(1

wl+1,m − wl+1,m+1+

1

wl ,m+1 − wl ,m

),

where αl and βm are parameter that only vary in l and m respectively.


Travelling Wave Reductions

Let us consider a travelling wave



0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10

Isolating the m = 15 and m = 20 cases, we see that

wl+2,m+5 = wl ,m,

so that after some time (in m), the wave has just translated in space.



0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10

In general, we will consider reductions of the form

wl+s1,m+s2 = wl ,m.

which we call a (s1, s2)-reduction.


We call a system of linear equations,



Ψl+1,m = Ll ,mΨl ,m




Ψl ,m+1 = Ml ,mΨl ,m





a Lax pair if the consistency






Ψl+1,m+1 = Ml+1,mLl ,mΨl ,m







Ψl+1,m+1 = Ll ,m+1Ml ,mΨl ,m








=⇒








=⇒ Ml+1,mLl ,m = Ll ,m+1Ml ,m









Implies our given lattice equation









Implies our given lattice equation

Q(wl ,m,wl+1,m,wl ,m+1,wl+1,m+1;αl , βm) = 0


The first example, the discrete modified Korteweg-de Vries equation

αl(wl ,mwl+1,m − wl ,m+1wl+1,m+1)

− βm(wl ,mwl ,m+1 − wl+1,mwl+1,m+1) = 0,

has the Lax pair

Ll ,m(αl/γ;wl ,m,wl+1,m) =

γ

αl

wl ,m

wl+1,m1

1γ

αl

wl+1,m

wl ,m

,

Ml ,m(βm/γ;wl ,m,wl ,m+1) =

γ

βm

wl ,m

wl ,m+11

1γ

βm

wl ,m+1

wl ,m

,

Equating Ll ,m+1Ml ,m with Ml+1,mLl ,m gives us the above.


The second example,the discrete Schwarzian Korteweg-de Vries equation

αl

(1

wl ,m+1 − wl+1,m+1+

1

wl+1,m − wl ,m

)= βm

(1

wl+1,m − wl+1,m+1+

1

wl ,m+1 − wl ,m

),

possesses a Lax representation where

Ll ,m(αl/γ;wl ,m,wl+1,m) =

1 wl ,m − wl+1,mαl

γ(wl ,m − wl+1,m)1

,

Ml ,m(βm/γ;wl ,m,wl ,m+1) =

1 wl ,m − wl ,m+1βm

γ(wl ,m − wl ,m+1)1

.

Equating Ll ,m+1Ml ,m with Ml+1,mLl ,m gives us the above.


Let us consider a (2, 2)-reduction, so that wl+2,m+2 = wl ,m.

We start with initial conditions in a staircase.




We label initial conditionsw0, w1, w2, w3,

with periodicity taken into account.






w3 w0

w1 w2

w3 w0

w1 w2

w3






w3 w0

w1 w2

w3 w0

w1 w2

w3

Recall our two equations,(1

wl,m+1−wl+1,m+1+ 1

wl+1,m−wl,m

)= βm

αl

(1

wl+1,m−wl+1,m+1+ 1

wl,m+1−wl,m

),

(wl ,mwl+1,m − wl ,m+1wl+1,m+1) = βmαl

(wl ,mwl ,m+1 − wl+1,mwl+1,m+1),

are functions of βm/αl alone.






w3 w0

w1 w2

w3 w0

w1 w2

w3

Recall our two equations,(1

wl,m+1−wl+1,m+1+ 1

wl+1,m−wl,m

)= βm

αl

(1

wl+1,m−wl+1,m+1+ 1

wl,m+1−wl,m

),

(wl ,mwl+1,m − wl ,m+1wl+1,m+1) = βmαl

(wl ,mwl ,m+1 − wl+1,mwl+1,m+1),

are functions of βm/αl alone.

Hence are reduction is consistent whenβm+2

βm=αl+2

αl:= q2,

where q is a constant.



w3 w0

w1 w2

w3 w0

w1 w2

w3

The second order difference equationsβm+2

βm=αl+2

αl:= q2,

are solved by

αl =

{a1q

l if l is odd,a2q

l if l is even,

βm =

{b1q

m if m is odd,b2q

m if m is even.



w3 w0

w1 w2

w3 w0

w1 w2

w3


βm=αl+2

αl:= q2,

are solved by

αl =

{a1q

l if l is odd,a2q

l if l is even,

βm =

{b1q

m if m is odd,b2q

m if m is even.

Hence, we need to consider m→ m + 2.



w3 w0

w1 w2

w3 w0

w1 w2

w3


βm=αl+2

αl:= q2,

are solved by

αl =

{a1q

l if l is odd,a2q

l if l is even,

βm =

{b1q

m if m is odd,b2q

m if m is even.


˜w0

˜w0

˜w2

˜w2

Our reductions are then given by

Q(w1,w2, ˜w0,w3; a1, b2qn) = 0,

Q(w3,w0, ˜w2,w1; a2, b1q2+n) = 0,

where n = m − l . We claim both reductions are q-PVI .



w3 w0

w1 w2

w3 w0

w1 w2

w3


βm=αl+2

αl:= q2,

are solved by

αl =

{a1q

l if l is odd,a2q

l if l is even,

βm =

{b1q

m if m is odd,b2q

m if m is even.


˜w0

˜w0˜w3

˜w1

˜w2

˜w2

Our reductions are then given by

Q(w1,w2, ˜w0,w3; a1, b2qn) = 0,

Q(w3,w0, ˜w2,w1; a2, b1q2+n) = 0,

Q( ˜w0,w3, ˜w1, ˜w2; a1, b1q2+n) = 0,

Q( ˜w2,w1, ˜w3, ˜w0; a1, b2q2+n) = 0,

where n = m − l . We claim both reductions are q-PVI .


w3 w0

w1 w2

w3 w0

w1

We first designate a spectral variable,


w3 w0

w1 w2

w3 w0

w1


x = ql

γ .


w3 w0

w1 w2

w3 w0

w1


x = ql

γ .

Our Ll ,m and Ml ,m were functions of x :

Ll ,m = L(a1,2ql/γ;wl ,m,wl+1,m),

Ml ,m = M(b1,2qm−lql/γ;wl ,m,wl+1,m),


w3 w0

w1 w2

w3 w0

w1


x = ql

γ .


Ll ,m = L(a1,2x ;wl ,m,wl+1,m)

Ml ,m = M(b1,2xt;wl ,m,wl+1,m),

where t = qm−l = qn.


w3 w0

w1 w2

w3 w0

w1


x = ql

γ .





We now construct operators equivalent to (l ,m)→ (l + 2,m + 2)

A(x , t) = L(a2xt;w2,w3)M(b2xt;w1,w2)

×L(a1x ;w0,w1)M(b1xt;w1,w2),


w3 w0

w1 w2

w3 w0

w1


x = ql

γ .





We now construct operators equivalent to (l ,m)→ (l + 2,m + 2)

A(x , t) = L(a2xt;w2,w3)M(b2xt;w1,w2)

×L(a1x ;w0,w1)M(b1xt;w1,w2),

and the operator equivalent to (l ,m)→ (l ,m + 2)

B(x , t) = M(b2xt;w2,w3)M(b1xt;w1, ˜w0).

This is a Lax pair for the non-autonomous reduction.


With the operators A(x , t) and B(x , t) equivalent ot shifts in l and m, theeffect of A(x , t) and B(x , t) are

Y (q2x , t) =A(x , t)Y (x , t),

Y (x , q2t) =B(x , t)Y (x , t),

hence, the compatibility in each of the cases is given by

B(q2x , t)A(x , t)− A(x , q2t)B(x , t).

For the case descibed, this gives the required reduction

Q(w1,w2, ˜w0,w3; a1, b2t) = 0, Q( ˜w0,w3, ˜w1, ˜w2; a1, b1q2t) = 0,

Q(w3,w0, ˜w2,w1; a2, b1q2t) = 0, Q( ˜w2,w1, ˜w3, ˜w0; a1, b2q

2t) = 0,

which is actually a consequence of the compatibility of Ll ,m and Ml ,m.


What are the properties of A(x , t)? Notice that in the mKdV case

det(Ll ,m) = det

γ

αl

wl ,m

wl+1,m1

1γ

αl

wl+1,m

wl ,m

=γ2

α2− 1,

Similarly for Ml ,m, hence, the determinant of A(x , t), as a product is

detA(x , t) = det L(a2xt)M(b2xt)L(a1x)M(b1xt),

=

(a2

1x2 − 1

) (a2

2x2 − 1

) (b2

1t2x2 − 1

) (b2

2t2x2 − 1

)a2

1a22b

21b

22t

4x8,

By a simple guage transformation, we may express this in the form

A(x , t) = A0 +A1

x2+

A2

x4.

where A0 is upper triangular with diagonal elements q2 and 1, and A2 islower triangular with diagonal elements q2/t2a1a2b1b2 and q2/t2a1a2b1b2.


By a transformation that diagonalizes A0, we may express A(x , t) in theform

A(x , t) =1

x4

((x2 − φ)(x2 − y) + ζ1 ω(x2 − y)

1

ω(x2ρ+ ψ) (x2 − ϕ)(x2 − y) + ζ2

)

where z1 =(a2

1y−1)(a22y−1)

b21b

22q

2t4z, z2 =

q2z(b21t

2y−1)(b22t

2y−1)a2

1a22

and

y = −tw1

(w4(a1w1+b1tw3)w2(a2w3+b2tw1) + 1

)w3

(a2b2w4(a1w1+b1tw3)w2(a2w3+b2tw1) + a1b1

) , z =

(t2 − a2

2y)

(a1b1w2y + tw0)

q2(1− b2

1y)

(a2b2w2y + tw0),

satisfies a version of q-PVI given by

zz =

(t2 − a2

1y) (

t2 − a22y)

q2(b2

1y − 1) (

b22y − 1

) , y y =q2(t2 − z

) (q2t2 − z

)(a1a2 − b1b2z) (a1a2 − b1b2q2z)

.

where y = ˜y = y(q2t),z = ˜z = z(q2t).


The reduction in the discrete Schwarzian KdV case is not as simple. Thefirst part is obvious, as

det Ll ,m = det

1 wl ,m − wl+1,mαl

γ(wl ,m − wl+1,m)1

= 1− α

x

hence the determinant of A(x , t) as a product is

detA(x , t) = det L(a2xt)M(b2xt)L(a1x)M(b1xt),

= (1− a1x)(1− a2x)(1− b1xt)(1− b2xt),

whereA(x , t) = A0 + A1x + A2x

2

but A0 = I and A2 is upertriangular with entries

θ1t = −b1b2t2 (w1 − w2) (w0 − w3)

(w0 − w1) (w2 − w3), θ2t = −a1a2 (w0 − w1) (w2 − w3)

(w1 − w2) (w0 − w3).


What is not so obvious is that the pair

θ1 = −b1b2t (w1 − w2) (w0 − w3)

(w0 − w1) (w2 − w3), θ2 = −a1a2 (w0 − w1) (w2 − w3)

t (w1 − w2) (w0 − w3).

are conserved quantities under the evolution equations

Q(w1,w2, ˜w0,w3; a1, b2t) = 0, Q( ˜w0,w3, ˜w1, ˜w2; a1, b1q2t) = 0,

Q(w3,w0, ˜w2,w1; a2, b1q2t) = 0, Q( ˜w2,w1, ˜w3, ˜w0; a1, b2q

2t) = 0,

such that θ1θ2 = a1a2b1b2. So we may parameterize A(x , t) as follows

A(x , t) =

tx (x − yn + z1) θ1 + 1 txδnωnθ2

tx(x − yn)θ1

ωntx (x − yn + z2) θ2 + 1

,

where z1 = (a1y−1)(a2y−1)z−1θ1ty

and z2 = (tyb1−1)(tyb2−1)−zθ2tyz

.


These y and z variables are given by

y =− ((w2 − w3)((w1 − w2)(a2(w1 − w0) + b1t(w0 − w3))

+ a1(w0 − w1)(w0 − w3)) + b2t(w0 − w1)(w1 − w2)(w0 − w3))

÷ (b1t(w1 − w2)(a2(w1 − w3)(w3 − w2) + b2t(w1 − w2)

(w0 − w3))− a1a2(w0 − w1)(w2 − w3)2),

z =− ((w0 − w3)(w3 − w2)(a1(w0 − w1)(w3 − w2)

− b2t(w0 − w1)(w1 − w2)− b1t(w1 − w3)(w1 − w2)))

÷ ((w0 − w1)(w1 − w2)((w2 − w3)(a1(w0 − w3)

+ a2(w3 − w1)) + b2t(w1 − w2)(w0 − w3)))

satisfies

zz =

(b1q

2ty − 1) (

b2q2ty − 1

)q2 (a1y − 1) (a2y − 1)

, y y =b1b2(z − 1)

(q2z − 1

)q2 (b1b2t − θ1z) (b1b2t − θ2z)


Notice that the parameterization

A(x , t) =

tx (x − yn + z1) θ1 + 1 txδnωnθ2

tx(x − yn)θ1

ωntx (x − yn + z2) θ2 + 1

,

under the determinantal condition

detA(x , t) = (1− a1x)(1− a2x)(1− b1xt)(1− b2xt),

is overdetermined. In fact, we obtain a curious non-autonomous necessarycondition,

V (y , z , t) =θ1(z − 1)2 − θ1y(z − 1) ((a1 + a2) z − t (b1 + b2))

+ y2 (a1a2z − θ1t) (θ1z − tb1b2) = 0,

In fact, the q-Painleve equation is determined by the equation

V (y , z , t)− V (y , q2z , q2t) = 0

V (y , q2z , q2t)− V (y , z , q2t) = 0,


-4 -2 0 2 4

-4

-2

0

2

4

Figure: 3 orbits of q-PVI on the pencil of biquadratics.Christopher M. Ormerod in collaboration with P H van der Kamp and G R W Quispel La Trobe University ()q-Painleve equations as periodic reductions of integrable lattice equations23rd of March, 2013 18 / 21

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4


-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4


-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4


-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4


-4 -2 0 2 4

-4

-2

0

2

4


-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4


Beyond period reductions

It is possible to extend this theory to reductions of the form

wl+s1,m+s2 = T (wl ,m)

where T is any symmetry of Q, i.e.,

Q(x , u, v , y ;αl , βm) = 0 =⇒ Q(Tx ,Tu,Tv ,Ty ;αl , βm).

For example, for discrete mKdV, we may apply the symmetry

wl+s1,m+s2 = Twl ,m = λwl ,m

and in the discrete Schwarzian KdV case

wl+s1,m+s2 = Twl ,m =λwl ,m + µ

δwl ,m + ε.

There is a method of obtaining Lax representations for these reductions.


Beyond period reductions

As one example, if we consider discrete potential KdV

(wl ,m − wl+1,m+1)(wl+1,m − wl ,m) = αl − βm

the reduction wl+2,m+1 = λ+ wl ,m, is consistent with the parameterization

wl ,m = (l −m)λ+ u2m−l = pλ+ un,

where we letαl = a0 + hl , βm = 2hm,

which gives us, for yn = un+2 − un+1, the equation

yn+1 + yn + yn−1 =a0 − hn − h

yn+ 2λ, (2)

and its Lax pair.


[1] B Grammaticos, A Ramani, J Satsuma, R Willox and A S Carstea,“Reductions of Integrable Lattices”, J. Nonlin. Math. Phys. 12,Suppl. 1, 363 (2005).

[2] M Hay, J Hietarinta, N Joshi, and F W Nijhoff, “A Lax pair for alattice modified KdV equation, reductions to q-Painleve equations andassociated Lax pairs”, . J. Phys. A, 40, Number 2, (2007), F61F73.

[3] C M Ormerod, “Reductions of lattice mKdV to q-PVI”, Phys. Lett.A, 376, Number 45, (2012), 2855–2859.

[4] C M Ormerod, P H van der Kamp, G R W Quispel, “On Laxrepresentations of reductions of integrable lattice equations”,arxiv:1209.4721.

[5] C M Ormerod, J Hietarinta, P H van der Kamp, G R W Quispel,“Twisted reductions of integrable lattice equations and their Laxrepresentations”. In preparation.


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q-Painlev e equations as periodic reductions of integrable ...cormerod/talks/WavesOrmerod2013.pdf · integrable lattice equations Christopher M. Ormerod in collaboration with P H